Skip to main content

Full text of "Supplementary algebra"

See other formats


D 
> 

(5 
(/» 
in 



CIA 
2\9 



UC-NRLF 



$B Sm Mfl5 




IN MEMORIAM 
FLORIAN CAJORl 




Heath's Mathematical Monographs | 

r, '' Mf-ral editorship of | - 

Webster Wells, S.B. 

: .;;^.ru- of Mathematics ■- -c Massachusetts Institute of Technology 



SUPPLEMENTARY 
ALGEBRA 



BY 



R. L. SHORT 




Heath h Co., Piibhs^^ 



New York Chicago 



Number lo Price^ Ten Cents 



The Ideal Geometry' 

Must have 

Clear and concise types of formal dem- 
onstration* 

Many safeguards against illogical and 
inaccurate proof. 

Numerous carefully graded, original 
problems. 

Must 

Afford ample opportunity for origin- 
ality of statement and phraseology 
without permitting inaccuracy* 

Call into play the inventive powers 
without opening the way for loose 
demonstration. 



Wells's Essentials of Geometry 
meets all of these demands. 



Half leathery Plane and Solid, J gQ pages. Price $1.2^^. 
Plane^ 'j^ cents. Solid, 75 ce;ns. 



D. C. H EATH & CO., Publishers 

BOSTON NEW YORK. CHICAGO 



STUDENTS IN ATTENDANCE AT STATE UNIVERSITIES OF 
THE CENTRAL WEST, 1885-1903. 





i 


s 


& 


s 


S 


1 


3 


s 


S 


s 


i 


s 


s 


g 


§ 


"1 


s 


s 


§ 




700 




































^>-/->^ 




700 


600 


































/ 






600 


600 


































/ 


/ 




500 


400 


































/ 


/ 




400 


300 
































/ 


/ 






»0 


200 




























li 




/ 


/ 




/ 200 


100 




























/ 




/ 






/ 


100 


3000 




























/ 




/ 






/ 


3000 


900 
























/ 








^ 






f 


900 


800 






















f 


/ 




/ 


" 






/ 




800 


700 






















/ 






/ 








/ 




700 


600 
















J 


-^ 


\. 


/ 






/ 








J 




600 


600 
















/ 










/ 








/' 


1 




600 


400 
















/ 








1 










■7 






400 


300 














y 










1 








/ 


/ 




/ 


800 


200 














/ 










/ 








// 


r 


-■' 




200 


100 














/ 








/ 










■y 








100 


2000 












/ 










/ 










// 








2000 


900 












/ 










/ 








.^ 


y 








900 


800 












/ 










/ 






/ 


/ 










800 


700 










/ 










/ 








// 


/ 










700 


800 










/ 










/ 






^'.-^ 


/ 


/ 






/ 




60O 


500 








/' 










/ 




/ 


x- 
><-. 


/ 










A. 




600 


400 






1^ 












/ 




// 






-/ 




J 


yi 




7 


400 


300 




y^ 












/ 






/ 







i. 




/ 


J 




/ 


800 


200 
















/ 


/ 


-h 




/ 




/ 






f 


/ 




200 


100 














/ 




/ , 


i 


> 


/ 




/ 


...... 


y J 


^7 


r 




100 


1000 














/ 


/ 


z 




/ 




J 


^ 






/ 






1000 


900 














/ 


A 


■r 






/ 


p, 
* 


y 


/ 










900 


800 












/ 


// 










i\ 






/ 










800 


700 










/ 


^ 


/ 


• 
J 




^^ 




/ 


^ 














700 


600 










J 


A 




^ 


-/) 


^ 




/ 
















600 


500 








/ 


+- 


/ 


/ 


/ 


J> 






















600 


400 






/ 


y 


rtSi 


■:^ 




A 
























400 


300 




—y 


^ 


::^' 








/ 
























800 




$ 


s 


£0 


88 


s 


1 


s 


8 


g 


c> 


s 


s 


s 


s 


s 


i 


s 


s 

1 


s 





Jllinoia 
J«ebruka>- 

'WUcoiuiiL- 
lowa. -^ 



Michigan 
Indiana - 
Kansas •— 

Minneiota . 
Ohio 



SUPPLEMENTARY 
ALGEBRA 



BY 

R. L. [short 



BOSTON, U.S.A. 
D. C. HEATH & CO., PUBLISHERS 

190S 



CAJORl 

Copyright, 1905, 
By D. C. Heath & Co. 



QA2, 

S55 



PREFACE 

The large number of requests coming from 
teachers for supplementary work in algebra, espe- 
cially such work as cannot be profitably intro- 
duced into a text, has led me to collect such 
material into a monograph, hoping by this means 
to furnish the teacher with methods and supple- 
mentary work by which he may brighten up the 
algebra review. 

As far as possible, illustrations have been drawn 
directly from calculus and mechanics, — this being 
especially true in the problems for reduction. 
Almost without exception, such algebraic forms 
are common to calculus work. For a large part 
of this list of algebraic forms in calculus, I am 
indebted to Miss Marion B. White, Instructor in 
Mathematics in the University of Illinois. 

No attempt has been made to demonstrate 
the theory hinted at in graphical work. Such 
treatment would involve a knowledge of higher 
mathematics. 

The graph of the growth of state universities 
was made by members of a freshman class of the 
University of Illinois. 

R. L. S. 



SUPPLEMENTARY ALGEBRA. 



Graphs. 



1. It is impossible to locate absolutely a point 
in a plane. All measurements are purely relative, 
and all positions in a plane or in space are likewise 
relative. Since a plane is infinite in length and 
infinite in breadth, it is necessary to have some 
fixed form from which one can take measurements. 
For this form, assumed fixed in a plane, Descartes 
(1596-1650) chose two intersecting lines as a co- 
ordinate system. Such a system of coordinates 
has since his time been called Cartesian. It will 
best suit our purpose to choose lines intersecting 
at right angles. ' 

2. The Point. If we take any point My its posi- 
tion is determined by the length of the lines QM= 
X and PM = y, the directions of which are paral- 
lel to the intersecting lines C^Xand (9F(Fig. i). 
The values x = a and y = b will thus determine a 
point. The unit of length can be arbitrarily chosen, 
but when once fixed remains the same throughout 
the problem under discussion. QM=x and PM 
=y, we call the coordinates of the point M. x, 
measured parallel to the line OX^ is called the 
abscissa, y, measured parallel to the Hne OY, is 

5 



6 Supplementary Algebra. 

the ordinate. OX and V are the coordinate axes. 
OX is the axis of x^ also called the axis of abscissas. 
(9 Fis the axis of y, also called the axis of ordinates. 
(9, the point of intersection, is called the origin. 






Fig. I. 



A plane has an infinity of points in its length, 
also an infinity of points in its breadth. The num- 
ber of points in a plane being thus, oo^, or twofold 
infinite, two measurements are necessary to locate 
a point in a plane. 

For example, x=2 holds for any point on the 
line AB (Fig. 2). But if in addition we demand 
that J = 3, the point is fully determined by the 
intersection of the lines AB and CD^ any point on 
CD satisfying the equation j?' = 3. 

3. The Line. Examine one of the simplest con- 
ditions in X and J, for example ;ir -j- j = 6. In this 
equation, when values are assigned to x^ we get a 



Graphs. 









Y 




A 




c 












D 














a' 
















X 












X 























Y ^ 




B 





Fig. 2. 



value of J/ for every such value of x. When x== o, 
j = 6;jr=i,j=S;;ir=2, j = 4;;ir=3, j=3; 
;ir = 5, J = I ; etc., giving an infinite number of 
values of x and j which satisfy the equation. There 
is, then, no definite solution. 

Laying off these values on a pair of axes, as 
shown in paragraph 2, we see that the points satis- 
fying this equation lie on the line AB (Fig. 3). It 
is readily seen that there might be confusion as to 
the direction from the origin in which the measure- 
ments should be taken. This is avoided by a 
simple convention in signs. Negative values of :r 
are measured to the left of the 7-axis, positive to 
the right. In like manner, negative values of j/ are 
measured downward from the ;r-axis, positive 
values upward. The regions XO F, VOX^, X'O Y\ 
Y^OXy are spoken of as the first, second, third, and 
fourth quadrants respectively. (See Fig. 2.) 



8 



Supplementary Algebra. 



By plotting other equations of the first degree in 
two variables (two unknown quantities), it will be 
seen that such an equation always represents a 




Fig. 3. 



straight line. This line AB (Fig. 3) is called the 
graph of X +j/ = 6 and is the locus of all the points 
satisfying that equation. 

4. Now plot two simultaneous equations of the 
first degree on the same axes, e.g. x-\-y = 6 and 
2x — 3^=— 3 (Fig. 4), and we see at once that 
the coordinates of the point of intersection have the 
same values as the x and y of the algebraic solution 
of the equations. 

This is, then, a geometric or graphical reason 
why there is but one solution to a pair of simulta- 
neous equations of the first degree in two unknowns. 
A simple algebraic proof will be given in the next 



Graphs, 9 

article. Hereafter an equation of the first degree 

in two variables will be spoken of as a linear 
equation. 





\ 




Y 


















\ 






















\ 












D 










\ 






,-/ 


y 














\ 


y 


A 














> 


? 


\ 














/ 


A 






\ 








J/ 


y 












\ 


.^ 


c 


^ 

















^5 














J 









Fig. 4. 

5. Algebraic Proof of Art. 4. Two simultaneous 
equations of the first degree cannot have two sets 
of values for x and y. Given the two equations 

ax-\-by=^c, (i) 

ex -{-fy = k, (2) 

Eliminating J/, {af — eh)x = cf — bh, (3) 

Let x^ and x^ be the roots of (3), different in 
value. Substituting these roots, we have 

{af— eb)x^ = cf—bky 

{of— eb^x^ = cf— bk, 

(of— eb){x^ — x^) - o. 



lo Supplementary Algebra. 

But x^^x^^y ,\af=eb, or - = — , which is im- 
possible. "^ 

In general, the plotting of two graphs on the 
same axes will determine all the real solutions 
of the two equations, each point of intersection of 
the graphs corresponding to a value of x and y 
satisfying both equations. 

6. It is well to introduce the subject of graphs 
by the use of concrete problems which depend on 
two conditions and which can be solved without 
mention of the word equation. 

Prof. F. E. Nipher, Washington University, St. 
Louis, proposes the following : 

"A person wishing a number of copies of a letter 
made, went to a typewriter and learned that the 
cost would be, for mimeograph work : 
$1.00 for 100 copies, 
1^2.00 for 200 copies, 
$3.00 for 300 copies, 
^.00 for 400 copies, and so on. 

" He then went to a printer and was made the 
following terms : 

$2.50 for 100 copies, 
;^3.oo for 200 copies, 
$3.50 for 300 copies, 

$4.00 for 400 copies, and so on, a rise of 50 
cents for each hundred. 

Plotting the data of (i) and (2) on the same 
axes, we have : 



(I) 



(2) 



Graphs 



II 




"The vertical axis being chosen for the price- 
units, the horizontal axis for the number of copies. 

"Any point on line (i) will determine the price 
for a certain number of mimeograph copies. Any 
point on line (2) determines the price and corre- 
sponding number of copies of printer's work.'* 

Numerous lessons can be drawn from this prob- 
lem. One is that for less than 400 copies, it is less 
expensive to patronize the mimeographer. For 400 
copies, it does not matter which party is patronized. 
For no copies from the mimeographer, one pays 
nothing. How about the cost of no copies from 
the printer } Why 1 

Any problem involving two related conditions, 
as in line (i), depends on an equation. 

The graph of an equation will always answer 
any question one wishes to propound concerning 
the conditions of the problem. 

The graph offers an excellent scheme for the 



12 



Supplementary Algebra. 



presentation of the solution of indeterminate 
equations for positive integers — a subject always 
hazy in the minds of beginners. 

Example. Solve 3;ir4-47=22 for positive inte- 
gers. Plotting the equation, we have 





Y 


















s 


V 






















N 














X 








^ 


s, 










o 










s 


s. 








2 

3 














k 




















k 




X 6 



















N 


s 























si 

4 
3i 

I 



Fig. 6. 

We see that the line crosses the corner of a 
square only when x= 2 and x=6. For all other 
integral values of ;r, y is fractional. The only posi- 
tive integral solutions are, therefore, x=2, 7 = 4; 
x = 6, j/=i. This corresponds to the algebraic 
result. 

7. The Curve. We will now plot a curve of 
higher degree than the first. Take, for example, 
;r2 -f j;/2 = 25, or ^2 _- 25 — ;r2. Giving x either posi- 
tive or negative values, will give positive and 
negative values for j/. When x=± i, 7 = ± V24 ; 
^=±2, j/ = ±V2i; x=±^yj/=±4; etc., until 
•^= ± S> after which value we find that y becomes 
imaginary. Plotting these values, we find that our 



Graphs. 



13 



equation is represented by a circle whose radius is 
5 (Fig. 7). 

Again, substituting values of x in x^-\-y^^2^y 
we have the sets of points in Fig. 7, which, when 
joined by a smooth curve, give a circle of radius 5 
for the locus of the equation, 
r 




X 


y 


±0 


±5 


±1 


±V24 


±2 


±V2T 


±3 


±4 


±4 


±3 


±5 


±0 



Fig. 7. 



2. 



Examples. 

s. 

6. 

3- ^= I. 

25 16 7. 

It will be noted that the graph of every equation 
in two variables of higher degree than the first is a 
curve. Solve by means of graphs, and compare 
results with the algebraic solution. 



X' 

25 16 

16 



+4=1. 



yl- 



16. 

y = x^ — x— 12. 



14 



Supplementary Algebra. 



















7^ 




JJ 


1 


t 


1- i - 


: t 


V t - 


r 


\ t - 




j^t -^ 





3 















































































X 


J' 


-3 


— 20 


— 2 


O 


— I 


6 


O 


4 


I 


o 


i 


-i 


2 


o 


3 


lO 



Fig. 8. 



Graphs. ir 

2. x^+J/^= 13, ;t:2~j^2_. j^^ 

It is seen in Example i that the line cuts the 
curve in two points. It is, in general, true that a 
straight line cuts a quadratic curve in two points 
only. These points may be real and different, real 
and coincident, or both imaginary. 

8. Examine the curve f = x^ — x^ — 4x + 4. Al- 
lowing X to vary, we obtain the set of values, 
x=- 3,7=-20; x=:-2yj/ = o; x=-i,j/ = 6; 
;r=o,j/ = 4, etc. (Fig. 8). 

Plotting our curve, we see that it has something 
of an inverted S-form, that it has two bends or 
turning points ; also, that it cuts the ;r-axis in three 
points; namely, when;ir= i, ;»r= — 2, ;r= 2. These 
are the roots of the equation just plotted. 

In general, there are as many roots to an equa- 
tion as there are units in the degree of the curve. 
Now these roots or intersections may not all be 
real. Take, for example, the curve j/ = x^-\-x^ — 
2;ir+ 12 (see Fig. 9). 

We see that the curve has the same general form 
as the curve in Fig. 6, but that it crosses the axis 
in only one point; namely, x= —3. An algebraic 
solution gives imaginary results for the other two 
roots, x= I + V— 3 and x= i — V— 3. That is, 
these two crossing points are imaginary. 



i6 



Supplementary Algebra. 



X 


y 


-4 


-28 


-3 


o 


— 2 


12 


— I 


14 


O 


12 


1 


"M 
"M 


I 


12 


2 


20 



H- 



Fig. 9. 



Graphs. 



17 



This shows also, to some extent, why imaginary 
points always go in pairs, for it is impossible to 
draw a curve so that it will cross the axis in only 
one imaginary point, or, in fact, in any odd number 
of imaginary points. 

Imaginary intersections may also occur in simul- 
taneous equations. In general, two simultaneous 
equations of the second degree intersect in four 
points, but the curves may be so situated that two, 
or perhaps all four, of the intersections may be 
imaginary. Take the curves. 



7- 



Sz 



7- 



k 



-N 



Fig. 10. 



Plotting these curves, we see that the first repre- 
sents a circle with its center at the point x= 2, 
y = 6, the radius of which is 2 ; the second repre- 



1 8 Supplementary Algebra. 

sents a circle having its center at the origin and 
a radius equal to 2 ; also that these circles do 
not intersect. Solving the equations algebraically, 



we get for the solutions, x = • 



5 



\ both of 



which are imaginary. (See Fig. lo.) The^-values 
are also imaginary. 

If we replace that first circle by 

(^~l)2 + (j- 1)2 = 4, 

we get real points for the algebraic solution, 

I ± V/ I T V/ 
X = , y = — 




Fig. II. 



Plotting the curves (Fig. ii), we see that there 
are also two real intersections. 

Note that the graphical method gives real inter- 
sections only. The reason is that our axes are 



Graphs. 



19 



chosen in a plane of real points, and imaginary 
points are in an entirely different field. Imagi- 
naries are plotted by what is known as the Argand 
diagram, but are not within the scope of the present 
paper. 

Example. Plot }-^=i, and x^-\'r^=i6 

25 9 
on the same pair of axes. The result should give 

four rea/ intersections. 



■ 

■ 

I ' I I I I 

T' 



Fig. 12. 

9. The Absolute Term. Determine what effect 
the absolute term has on the curve. Examine the 
curve j/=x^—x—i2 (Fig. 12). The curve turns at 
the point x=^y j/= — 12^, and when;ir=o, j^= — 12. 
Replace 12 by another constant, say 6. The curve 
now becomes j^=x^—x—6 (Fig. 13), which turns 



20 



Supplementary Algebra. 




at the point x=^y j^ = — 6|, and crosses the 
j^-axis at —6. Replace 6 by o. y^x^-'X — o 
(Fig. 14). This curve turns at ;r = |, 7 = — J, and 
crosses the axes at the origin. 




Fig. 14. 



Graphs. 



21 



In these three figures the curves are of the same 
form, but are situated differently with respect to 
the axes. It is seen, then, that the absolute term 
has to do with the position of the curve, but has 
nothing to do with its form. Moreover, a change 
in the absolute term lowers or raises the curve, 
and does not shift it to the right or left. 









1 


i 














t 








1 


\ 

















X 






I 
I 












/ 














1 

I 












/ 
1 














\ 












1 




1 










\ 

\ 












1 




1 










\ 

1 












1 

< 














1 


1 


























\ 








1 


















\ 

\ 








/ 


















\ 








/ 




















\ 




\J 




J 






















X 




1 


















' 


V 




V 


























r 










■ 



Fig. 15. 

ID. Change in the Size of Roots. We learn in 
the theory of equations that the substitution of 
x-\-n for Xy where n is positive, gives another 
equation, the roots of which are each less by n 
than the roots of the original equation. Or, if 
;r — ;^ be substituted for ;r, where n is positive, each 
root of the new equation will be greater by n than 



22 Supplementary Algebra, 

the roots of the original equation. In 4: = 6, if 
;r + 2 is substituted for ;r, we get x= 4, 3. root less 
by 2 than the former equation. 

Plot the curve y = x^ — x — 12 (Fig. 1 5), the con- 
tinuous curve. Substitute x-\- 2 for x in the above 
equation, and we have j/ = (x+ 2j^ — {x+ 2)— 12, 
or7 = ;r2 + 3;i;— 10; plot on the same axes — the 
dotted curve. The transformed curve crosses the 
axis at x=2f x== — S, while the original curve 
crosses at -;ir = 4, ;ir= — 3, the new roots each being 
2 less than the old ones, while the form of the 
curve is unchanged. 

We see, then, that a substitution of x±n for x 
shifts the curve, parallel to itself, to the right or 
left, and does not change its form. The shape of 
the curve must therefore depend on the coefficients 
and the exponents of x and j/. 

Plot f = x^, y=z(^x—if, y=={x—2fy on the 
same axes. It is seen that the curves are the 
same curve, merely shifted each time to the left. 
All curves of the form j/ = (x — of can be reduced 
to the form j/ = x^hy shifting the axes. 

Plot J/ = x^, J/ = x^, J/ = x^, y = ;tr2«, on the same 
axes where a is positive, and note the similarity 
(but not an identity) in the form of the curves. 

II. Plotjv = - for both positive and negative 

values of x. We see that as x grows smaller, y 
constantly increases, and that as x approaches 
zero, y becomes infinite. When x is positive, the 



Graphs. 



23 



curve lies wholly in the first quadrant, approaching 
positive infinity on the ;ir-axes when y is very small, 
and positive infinity on the ^-axes when x is very 
small (Fig. 16). When x is negative, the curve is 
of the same form and in the third quadrant. 

Plot the curve y=^—^' Note that if a plane 
x^ 

mirror were passed through the ^r-axis perpendicu- 
lar to the j-axis, the reflection of the curve y ^—^ 

I ^ 
in the mirror would be the graph oi y= -• 



Fig. 16. 

It has been seen in paragraph 10 that the curves 
yz=ix^^ etc., touched the ;r-axis, but did not cross 
it. These curves had equal roots (when jv = o, 
(jt — i)2 = o), which give .r = i or i. It is in gen- 
eral true that a curve with equal roots must be 
tangent to the axis, i,e, must cut the axis in at 



24 Supplementary Algebra. 

least two ^consecutive points. If the equation has 
in its numerator a binomial factor (x — a) repeated 
an even number of times, the curve will touch the 
axis at a, but will not cross at that point. But if 
the factor be repeated an odd number of times, 
the curve will touch the axis and cross it also at a 
[try J = (-^ — I /] . If the binomial is not repeated, 
the curve crosses the axis at a non-vanishing angle. 
All curves where the factor repeats are parallel to 
the axis at the point of intersection. This fact 
is at once apparent when we remember that the 
curve is a tangent to the axis at such a point. 

12. Maxima and Minima. It may be noticed 
that many of these curves make a complete turn, 
as in Fig. 8 or Fig. 1 2 ; that is, there is some point 
which is higher or lower than any other point of 
the curve in the immediate vicinity of the point in 
question. Such turning points are called maximum 
and minimum points of the function. These points 
are of considerable geometric value. 

Problem. Suppose it be required to find the 
rectangle of greatest area which can be inscribed in 
an isosceles triangle with an altitude equal to 2 and 
a base equal to 2. Let ABE (Fig. 17) be the given 
triangle, DCy one half the required rectangle. Let 
J be the area of the rectangle and 2 x the base. 

* Mathematically the word coincident should replace consecutive. 
The beginner, however, seems more clearly to realize that the 
intersections are approaching each other indefinitely near if consec- 
utive is used. 



Graphs. 



25 










~"~~ 


X 








0] 


























































































































Y 




\ 1 





Fig. 18. 

From the figure, 

CO : AD-OD : : 2 : i, 

C0^2{AD-0D\ 

= 2(i-;r), 

therefore y^^x — A^x^ (Fig. 18). But this ex- 
pression is a curve where y represents the area we 
are trying to find, and may be plotted as any other 
curve. 

Plotting the curve, we see that it turns at ;r = |-, 
j^= I (Fig. 18), and that this point is higher than 
any other point of the curve. 

Therefore, the greatest rectangle which can be 
inscribed in the given triangle is one in which the 
base and altitude are each i. The rectangle is 
minimum when x=o and when x= i. Why ? 

Again : Given a square piece of sheet metal 30 
inches on a side, find the side of a square to be 



26 



Supplementary Algebra. 



cut out of each corner of the sheet so that the 
remainder will fold up into a box of maximum 
volume. 

Let ABCD (Fig. 19) be the sheet of metal and 
y the volume of the box. Let x be the side of the 
square to be cut from each corner. 









s 

7 














H^ 




Fig. 19. 



Fig. 20. 



Then from the figure, y = (30 — 2 xfx. 
Plotting this curve (Fig. 20) we see 
that it turns twice, once at ;r = 5 and 
once at x= 1$; also that ;r= 15 is at 
the lowest part of a bend, while ;r= 5 is 
at the highest part of the other bend. 
This shows, then, that the box is greatest 
when ;r= 5 and least when ;r= 15, for 
when 15 is taken from each corner no 
material remains from which to make 
the box. The graph shows also that 



X 


I 








2 


1352 


5 


2000 


6 


1944 


8 


1568 


10 


1000 


13 


208 


14 


56 


15 





16 


64 



Graphs. 27 

the greatest box that can be so formed has a 
volume of 2000, the value of y of the curve, when 

The turning points of a curve are often difficult 
to plot because the values of x and y must be taken 
so closely together. Much labor may often be 
saved by the following process, called differentia- 
tion. Only one rule is given here. Should the 
reader wish rules governing all cases, he will find 
them in Wells's ** College Algebra," p. 472, and 
Wells's "Advanced Course in Algebra," p. 527. 

The following rule holds where j/ equals a rational 
polynomial in Xy containing no fractions with vari- 
ables in the denominator : multiply each coefficient 
by the exponent of x in that term and depress the 
exponent by one. If the resulting expression is 
equated to zero and solved for x, the roots thus 
obtained will be the abscissas of the points where 
the curve turns. A proof of this will have to be 
postponed until higher mathematics is reached. 

Illustration. Use the curve in Fig. 8. 

j^ = ;r3-;tr2-4;r + 4. (l) 

Differentiating and equating the result to zero, 
we have ^:^-2x~4 = 0, (2) 

whence x=— — ^= 1.5+ or — ,^6+y 

3 
the turning points of the curve. This result may 

be verified by examining Fig. 18. These two 



28 



Supplementary Algebra. 




Fig. 21. 



Graphs. 29 

curves are plotted on the same axes in Fig. 21, 
equation (2) being dotted. 

Problem. An open vessel is to be constructed 
in the form of a rectangular parallelopiped with a 
square base, capable of containing 4 cubic inches. 
What must be the dimensions to require the least 
amount of material ? Solve by plotting. 

Ans. Dimensions must be 2 x 2 x i inches. 

It is often necessary to plot a curve in one vari- 
able only; for example, ;ir2—;r— 12 = o. In such 
a case we are really seeking the intersection of two 
curves, j/ = o and x^—x— 12 = 0, which are plotted 
as above, y = o being the Hne OX. In such a case 
the curve is readily plotted if the two equations are 
equated ; e,g, y^x'^—x— 12. 

Flot x^ — X — 6 = o; also regard the equation as 
of the form ax^ -\- bx -{- c = o, and solve by the 

formula, x = . Note the nature of 

2a 

the radical part of the formula after the substitution 
is made. 

Treat x^— x-{-^==o and x^—x-{-6 = o in the 
same manner. Remember that the places where 
the curve crosses the ;ir-axis correspond exactly with 
the algebraic root obtained. This affords an ex- 
cellent method for illustrating to the student the 
meaning of imaginary roots. 



30 Supplementary Algebra. 

Short Methods. 

(To shorten the work in certain classes of fractions involving 
addition. For review work only.) 

Rule I. If two fractional numbers having a 
common numerator and denominators prime to 
each other are to be added, multiply the sum of 
the denominators by the common numerator for 
the numerator of the result; the product of the 
denominators will be the denominator of the result. 

Rule II. If two fractional numbers having a 
common numerator and the denominators prime 
to each other are to be subtracted, subtract the 
first denominator from the second, multiply this 
difference of the denominators by the common 
numerator for the numerator of the result; the 
product of the denominators will be the denomi- 
nator of the result. 

Illustration : 

i + * = T^; f + l = 2G + i) = if- 
4-i=A; (f-l) = 2(J-i) = ^5- 

This scheme makes possible a great saving of 
products in algebraic summations. 



+ ■ 



(use Rule II), 



Short Methods. 31 

_ 4 , -4 ^J I i_\ 

(use Rule II), 
12 



-"(^4-4X^4-1) 

2. — "^ ^=-J! —' (Use Rule II.) 

X—l X—2 ^—3 X—4 

— I I . 

(x-i){x-2)'~'^{x-3)(x-4y 
whence 

(^-3)(^-4) = (-^-"i)(^-2); 
:ir2— 7;ir+ 12 = ;ir2 — 3;ir4- 2. 
.-. ;r=2l 

3. 7= H 7== = 12 : solve for x. 

X + V ;r2 — I X — -Vx^ — I 

( , /% + 7t=V ^2 (use Rule I); 

\x + V;!:^ — I X — -yx"^ — 1/ 

24r 

12. 



;r2 — (;r2— i) 

Improper Fractions. 

The larger the factor, the more the multiplication 
involved, the greater the liability of error, and the 
slower the speed. The times demand, in mathe- 
matical computations, accuracy combined with a 
reasonable degree of rapidity. 



32 Supplementary Algebra. 

In general, improper fractions have no place in 
problems involving addition and subtraction. 

In arithmetic we do not add |, f , f , etc., because 
the work is cumbersome and the factors large. 
We reduce these fractions to whole or mixed num- 
bers, 2f, 2 J, if, add the integers, and then the 
fractions. We should not think of adding the 
fractions in their original form. The same prac- 
tice holds good for algebra. An improper fraction 
may be defined as a fraction in which the numer- 
ator is of the same or higher degree than the 
denominator, both being rational. 

Examples. 

X—l x— 2 _ x — 3 X — ^ 
X—2 X — 2i ^—4 ^-"5 
Divide each numerator by its denominator — 
parenthesis division. 

I I.I I 

H I -.= iH ^- i_. 



X--2 x—i ^—4 ^—5 

Collecting, 

\_X-2 X-l\ \_X-\ X-<>\ 

(Use Rule II.) 

—I _ — I 

(Ar-2)(;r-3)~(;tr-4)(^-5)' 
{x-^)(x-l) = (x-2){x-z). 



Short Methods. ^3 

Such reduction of improper fractions will in most 
cases shorten a solution and will reduce the size of 
the factors ; a binomial numerator generally reduces 
to a monomial, and a trinomial numerator to a 
binomial. 

2. ^ ^ ^ 



^ + 7^+ lO X — 2 

x^-{- yx-\- lo 



Reducing, i + o^,^^,^ - i 



Collecting, factoring first denominator, 

7 r-r^^ r bCCOmOS ; ]• 

{x -{- i){x -\- 2) X—2 \x-\-2 X—2j' 

(Use Rule II.) 

X^—X-^- I . X^-\-X'\- I 

3. 1 ; ^2X. 

X— I X-\' I 

Reducing, x-^- — 7 + -^ + rT:7 = 2;r, 
X — 1 X "i" 1 

whence, 1 ; — = o. (Apply Rule I.) 

X—l X-\-l \ irtr J / 

r.x — o. 

Problems of the above type are common to all 
texts on algebra. After the student has solved 
them by the usual methods, so as to become fa- 
miliar with the principles involved, he should be 
given a review where these briefer solutions might 
be introduced. 



34 Supplementary Algebra. 

Note. A short method should never be introduced early 
in a subject. Such procedure clouds the pupil's ideas of mathe- 
matical principles. 

Simultaneous Quadratics of Homogeneous Form 
may often be solved as follows : 

Illustrative Problems. 



Eliminate the abso- 
lute terms by multi- 
plying equation ( i ) by 
(c) 9f equation (2) by 14, 
and subtracting. 

This resulting func- 
tion of (;r,^)is always 
capable of being fac- 
tored, either by in- 
spection or by quad- 
ratic factoring and 
gives two values of x 
in terms of j. These, 
when substituted in 
( I ) or (2) will produce 
the required values 
for J/. X is then 
easily obtained. This 
method avoids the 
substitution of y'=^vx. 



2x^ — xy^2S 

;i:2 + 2/= 18 


(I) 
(2) 


i%x^ — gxy = 2S2 

l4;r2 + 2872=252 


(3) 
(4) 


4;r2 — ()xy — 2872 = 


(5) 


(4^+7j/)(^~4J^) = 


0. 


,\x=-lyor 4y. 




Substituting in (2), 




11/ + 2/ =18. 




^^^^-18. 

16 




y'=¥- 




.•.J/=±|V2. 


16/ +2/= 18. 




y=i. 





± I. 



Short Methods. 35 

Algebraic Problems. 

The following problems in reduction are of com- 
mon occurrence in the higher mathematics, and 
especially in the calculus, where long lists of such 
forms are found in the work almost daily. They 
appear most frequently in differentiation and prep- 
aration for integration ; lack of facility in handling 
such forms handicaps a pupil and makes him think 
the calculus a difficult subject. 

Type Forms. 

(a + df -{-(a-df=2 (a^ + 6^) ; 

{a + df-4ad = (a'-df; 
{a-df-h4ab = {a + df. 

It is desirable that a student be so familiar with 
these type forms that he may, by inspection, write 
down the results of similar operations. 

a -]- d a — b 



(7 + V25 -/)2 - (7 - V25 -/)2 = .? 

Simplify the following : 
{e^'he-^f-^4, 



^6 Supplementary Algebra. 

■\/(mn — i)2 + 4 mfif 

Negative Exponents. 

Notice all quantities which have negative expo- 
nents. Multiply both numerator and denominator 
of the given expression by the product of all such 
quantities with the sign of the exponents changed. 
If the expression be integral, regard its denomina- 
tor as unity. 

Example (i): 

Multiply both numerator and denominator by 

2 

Result = ^ ^ 



Example (2) : 

x(i+x^y^-x{i-x^y^. 



•y/l — X^\I -\- X^ 

Multiplying by , we obtain 

Vl —x'^^Y ■\-X^ 
x^i —x'^ — x^i '\-X^ 



Short Methods. 37 



In like manner simplify : 






,-.+^^ fif+f-^-* 



jrJj/* 



In all reductions watch for opportunities to 
factor, especially to remove factors common to 
several terms. 

(«) Z{x - 2f{2 X -^ if -^ t,{2 X ^ \)\x - 2f 

= {2X+ lf(x-2f(l8x- II). 

(b) 3{x-i)\x + 2f + 4(x-if{x+2f 

= {x-if{x + 2f(7x+s). 

(c) (a^+x^f- 3 x^(a^+ x^f = y/'^V^{a^- 2 :fi). 
(^) 8;r(4;r2 - 3)(;r2 + l)*+ 24;r(;^ + I)* 

= 56;t8(;tr2+i)*. 

(^) |(;r+l)-^(;p-5)2 + 2(x-5)(;ir+l)* 

= |(;r+l)-i(^-5)(2;r-l). 

Problems. 



1. Reduce == to the form 



Vi-;.^ i-^ 



38 Supplementary Algebra. 



a 



2. Reduce s-f^^ -- 

a^ x-\-a 

x^ x — a 

3. Reduce 



4. Reduce 

I 



V^ 



to the form 



ax — x'' 



■H^-i^' 



5. Reduce 

m _m m jm 

to the form 



X'^—X "" x^-\-x~^ 

6. Reduce 



' + ■\/x^ — cfi x^ jx^ — a 



a ^ d^ 



to the form -\ 

X ^ x — a 



7. Reduce 
to unity. 



Short Methods. 



39 



8. Reduce 
ax 



{a^ - x^)i 






to the form 



V^2z:^ 



9. S^VWK\ If K^ ^^''y-y\ find 
/ — -^ 

10. Reduce 2 
I 

Vl — 3;ir — ;tr2 

11. Reduce 



to the form 



V13 



:x^-x^-6 



to the form — 



— 4;^^ 

25 I A-2^ y 



12. Reduce V2 ax — ;r2 to the form 
a {x — df 



13. 2 tan ;ir 4-(tan ;i:)2 — 3 = o ; find tan ;ir. 



14. 2 cos ;r H = 3 ; find cos x. 

cos;r 



40 Supplementary Algebra. 



15. h2 cot;r=fVi -f-cot2;r; find cot ;r. 

cot;r 

16. Reduce ; r-^ to the form 

L_ + I 4 

2(l+-sr) I0(l--8r) 5(3 4- 2-8') 

17. vS = — ^(i2 + xy - 8. Evaluate 5 when 

18. Reduce 



M 4^ V 

V2V1-W 



2x— 2;rVi —x^ 



to the form -li-\ — V 

19. Reduce 

_ . 2;r— I 



■y/x^ —x— I 



to the form 



2;r— I + 2 ^x^—x— I 

I 



V^^^-jF— I 
20. Reduce 



/£~2\i _^ 3 fx^2\'\f 2 Y 

V;y+2/ 4V;tr+2/ \:rH-2/ 

W+2/ 



+ 2/ 
;i^ — I 

to the form —^ 



Short Methods. 


21. Reduce 


[-(^)i^^ 


X _x 

ea + e a 


2a 


to the form «(^^-^"^X 

4 


22. Reduce 


— r 
f 

to the form —y. 


23. Reduce 


X :i— 

-4/ 


,2 



41 



y 



to 3-^ + 2/ when y = 4/;r. 

24. Reduce 



to 






-<=m 



a^L 



jr2 -1/2 
when ±~-i^2L=i, 



42 Supplementary Algebra. 

25. Reduce 

V «VA ah I 



^ --^ 






26. Reduce 



^-T/i/ 3 / i/*\ 'i/^;tr'~^ 






to 



^s 3 ;r^^ 



when x^ -hj/^ = ^^. 



27. Reduce 

„i\2ll 



1 to — ^^ — Y^ when x^ -{-y^ = a^. 



£^ 

2X 

28. Reduce 






2a 



1/2 ^ ? _? 

to — when ^=:-(^ + ^ «). 



Short Methods. 43 

29. Reduce 



to 



2V;r 

{2 a — xf 

'y/x{2a — xf 
30. Reduce 






31. ^-^v^:=^= ^^l-"^V -^i), (I) 



^ = r— ^-^^ (2) 



^1 



^1 

Eliminate ;irj between (i) and (2) and get 
x'^^-y^^ia-Vbf. 

32. Reduce 

\^x-a^x-{-a — \{x'\-df{x — aJ^ 
X — a 

(x— 2 (i\\/x 4- d 
to ^^ — 3 

{x-^df 



44 Supplementary Algebra. 

33. Reduce 

n(i'\'xY * x""-^ -nii-V xY-^ • x'^ nx"^"^ 
^ to 



34. Reduce 

2(i-a^)-{-4x^ 

35. Reduce ^ ^-r^r- to 



-(tST ■""■ 



36. Reduce 






37. Reduce 
;.(l+^)-^-;t-(l-^)-^ to ifi - ^ 1 . 



38. Reduce 



(l-2;ir'^)^ ^ If 4;tr-2 



j^, -" ,^ 2L2^-2;tr+I 



Vl - 2 W 

— 4^±?— Ito-^. 
2;r2-h2;r+iJ 1+4^ 



Short Methods. 



45 



39. Given 



show that (a+/9)*+(a~/3)*=2>^*. 



40. Given 

a^ (k-af 
a^ {k-af 



; show that x^ +7' = /^^ 



Wells's Mathematical Series* 

ALGEBRA. 
Wells's Essentials of Algebra . • • . , $z.zo 

A new Algebra for secondary schools. The method of presenting the fundamen- 
tal topics is more logical than that usually followed. The superiority of the 
book also appears in its definitions, in the demonstrations and proofs of gen- 
eral laws, m the arrangement of topics, and in its abundance of examples. 

Wells's New Higher Algebra . . . . .1.3a 

The first part of this book is identical with the author's Essentials of Algebra. 
To this there are added chapters upon advanced topics adequate in scope and 
difficulty to meet the maximum requirement in elementary algebra. 

Wells's Academic Algebra . . . • , x.o8 

This popular Algebra contains an abundance of carefully selected problems. 

W^ells's Higher Algebra ...... 1.32 

The first half of this book is identical with the corresponding pages of the Aca- 
demic Algebra. The latter half treats more advanced topics. 

Wells's College Algebra ...... 1.50 

Part II, beginning with Quadratic Equations, bound separately. $1.32. 

Wells's Advanced Course in Algebra .... $1.5^ 

A modern and rigorous text-book for colleges and scientific schools. This is the 
latest and most advanced book in the Wells's series of Algebra. 

Wells's University Algebra . . • . .1.3a 

GEOMETRY. 

Wells's Essentials of Geometry — Plane, 75 cts.; Solid, 75 cts.; 

Plane and Solid . . . . . . . z.25 

This new text offers a practical combination of more desirable qualities than 
any other Geometry ever published. 

Wells's Stereoscopic Views of Solid Geometry Figures • .60 

Ninety-six cards in manila case. 
Wells's Elements of Geometry — Revised 1894. — Plane, 75 cts.; 

Solid, 75 cts.; Plane and Solid . . • • • z.as 

TRIGONOMETRY. 
W^ells's New Plane and Spherical Trigonometry • . $1.00 

For colleges and technical schools. With Wells's New Six-Place Tables, $1.25. 

Wells's Complete Trigonometry . . . , . .90 

Plane and Spherical. The chapters on plane Trigonometry are identical with 
those of the book described above. With Tables, $1.08. 

Wells's New Plane Trigonometry • • • • .60 

Being Chapters I-VIII of Wells's Complete Trigonometry. With Tables, 75 cts. 

Wells's New Six-Place Logarithmic Tables • • • .60 
The handsomest tables in print. Large Page. 

Wells's Four-Place Tables . • • • • .35 

ARITHMETIC. 
Wells's Academic Arithmetic • • • • • $1.00 

Correspondence regarding terms for introduction 
and exchange is cordially invited. 

D. C. Heath & Co., Publishers, Boston, New York, Chicago 



Wells's 
Essentials of Geometry 

DEVELOPS SKILL IN 

THE BEST METHOD OF ATTACK 

IN THE FOLLOWING WAYS 



It shows the piipii at once the parts of a proposition and the 
nature of a proof. Sections 36, 39, 40. 

11. 

It iat-'-n 'yrv- ■■■n ;■;;.;.•::! ::,■.;<;;-,;;,-, " •^,.:■H', and gives such aid as 
will lead the student to see how each proof depends upoa some- 
thing already given. See pages 17-18. 

HI. 

It begins very early to leave something in the proof for the 
student to do. See section 51 and throughout the book. 

The converse propositions are generally left to the student. The 
'-' indirect method '' is used in those first proved and thus gives a hint 
as to 1k)\v such propositions are disposed of. See page 21. 

V. 

.^ Uj:^o :i;:iiu.cr of the simpler propositions, even iu the first book, 
are left to the student with but a hint. See pages 50, 51, 52, etc. 

VL 
iiguics with all auxiliary lines are made ilu e • i^ 

the student learns why and how they are drawn. 

VIL 
Atter Book I the authority for a statement m the proot is r<ot 
stated, but the quesdon mark used or the section given. 



D. C. HEATH & CO., Publishers 

BOSTON NEW YORK CHICAGO 




MATHEMATICAL MONOGRAPHS 

ir:SU}-I> UNDER THE GENERAL EDITORSHIP OF 

WEBSTER WELLS, S.B. 

Profiwmr of Matluyncdl a in the ^fai::■'■■■ / ;/.: :'■.; Insiiiide of Technology. 



It is the purpose of this series to make direct contribution 
to the resources of teachers of mathematics, by presenting 
freshly written and interesting monographs upon the history, 
theory, subject-matter, and methods of teaching both elemen- 
tary and advanced topics. The first numbers are as follows : — 

1. FAMOUS GEOMETRICAL THEOREMS AND PROBLEMS AND 

THEIR HISTORY. By William W. Rupert, C.E. 

u TJie Greek Geometers, iu The Pythagorean Proposition. 

2. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert. 

ii. The Pythagorean Proposition (concluded), iii. Squaring the Circle.' 

3. FAMOUS GEOMETRICAL THEOREMS. By W^illlam W. RLtptRT. 

jv. TnsrctK vn vA au Angle. Y. The Area of a Triangle in Ternis of its Sides. 

4. FAMOUS GEOMETRICAL THEOREMS. By William W, Rupert. 

vj. The Duplication of the Cube. vii. Mathematical. Inscription upon the 

'i"i>i.i. u.,;r: of LiKlolph Van Ccnlcn. 

5. OJ^ TEACHING GEOMETRY. By Florence Milner. 

6. GRAPHS. By rrofr-ssor R. J. Ally, Indiana University. 

7. FACXORIK^G. By l^ofessor W,i^:b.ster Wells. 

PRICE, lo CENTS EACH 



D. C HEATIi & CO., Publishers, Boston. New York. Chicago 



(GAyLAMOUNrl 

PAMPHUT BINOEk 

Manufaclutid by 

;gaylordbros.Uc.3 

Syracus*, N. Y. 
Stockton, Calif. 



__~^^j^ojectto immediate recall. 



"PR 5 msm 



27Nov'57GC 



lOV IS flE7 

22fab'60J(J 
REC'D CD 

2l-100m-l,'54(1887sl6) 



476